Long term behavior of solutions of the Lotka-Volterra system under small random perturbations.
type: Item
instance of: scholarly article
title: Long term behavior of solutions of the Lotka-Volterra system under small random perturbations.
zbMATH Open document ID: 1061.34513
author: Fima Klebaner
author: R. Z. Khas'minskiĭ
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: This paper deals with a stochastic analogue of the Lotka-Volterra model for predator-prey systems where the birth rate of the prey and the death rate of the predator are perturbed by independent white noises with intensities of order \(\varepsilon^2\) (\(\varepsilon\) is a small parameter). Such a model can be described by the stochastic differential equations \[ dX_t=X_t(a-bY_t)dt + \varepsilon \sigma_1 X_tdW_t^1,\quad dY_t=Y_t(-c+dX_t)dt + \varepsilon \sigma_2 Y_tdW_t^2, \tag \(*\) \] where \(X\) and \(Y\) are the population sizes of prey and pradator, respectively, \(a,\,b,\,c,\,d,\,\sigma_1,\, \sigma_2\) are positive constants, and the stochastic perturbations \(W_t^1\), \(W_t^2\) may be either of Itô or Stratonovich type. The authors show that for small initial population sizes such stochastic model is adequate, whereas for large initial population sizes it seems to be not so suitable, because it leads to ever increasing fluctuations in population sizes, although it still precludes extinction. In particular, the authors establish that large population sizes of predator and prey coexist only for a very short time, and most of the time, one of the populations is exponentially small. This paper is a continuation of previous investigations of \textit{R. Z. Hasminskij} [Kybernetica, Praha 4, 260--279 (1968; Zbl 0231.60045)] and extends also results by \textit{L. Arnold, W. Horsthemke} and \textit{J. W. Stuckl} [Biom. J. 21, 451--471 (1979; Zbl 0433.92019)].
reviewed by: Józef Myjak
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1906241
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: predator-prey coexistence
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015345354
OpenAlex ID: W2054615472
Wikidata QID: Q57712795
cites work: The Influence of External Real and White Noise on the L<scp>OTKA</scp>‐V<scp>OLTERRA</scp> Model
cites work: Q5653395
cites work: Q3923307
cites work: Q5639096
cites work: Q4236350
cites work: Thermodynamics, real and complex periods of the Volterra model
cites work: The period in the Lotka-Volterra system is monotonic
DOI: 10.1214/AOAP/1015345354
Recommended article: Influence of stochastic perturbation on prey-predator systems
Recommended article: Persistence and stationary distribution of a stochastic predator-prey model under regime switching
Recommended article: Asymptotic behavior of a non-autonomous predator-prey model with Hassell-Varley type functional response and random perturbation
Recommended article: Q5101628
Recommended article: Q5290190
Recommended article: Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator
Recommended article: Some results on the Lotka--Volterra model and its small random perturbations
Recommended article: Asymptotic analysis and extinction in a stochastic Lotka-Volterra model
description: scientific article; zbMATH DE number 1906241